Alternative Tests of the Expectations Hypothesis of the Term Structure of Interest Rates

Technical Paper

2/RT/01

Jan.

2001

Alternative Tests of the Expectations

Hypothesis of the Term Structure of

Interest Rates

By

Don Bredin

∗∗∗∗

Central Bank of Ireland

The views expressed in this paper are the personal responsibility of the author and are not necessarily held either by the Central Bank of Ireland or by the ESCB. The author would like to thank Keith Cuthbertson, Stuart Hyde and Dirk Nitzsche for helpful comments. All errors and omissions are the authors’. Comments are welcome.

∗ _{Corresponding Author: Don Bredin, Economic Analysis, Research and Publications Department,}

Central Bank of Ireland, PO Box 559, Dublin 2.

Abstract

This paper reports the results for a number of alternative tests of the term structure of interest rates using Monte Carlo analysis. The study provides an extension to previous work done on the term structure using Irish money market rates. The aim of the paper is to provide evidence that rejections of the expectations hypothesis may be due to inaccurate statistical testing rather than the data being inconsistent with the theory. The study first focuses on some traditional methods of testing the EH using single equations. Then some alternative tests, which are frequently used in the literature, are discussed. However, we show using Monte Carlo experiments that the rejection of the EH using these alternative tests may be due to poor statistical properties in finite samples rather than a rejection of the theory.

1. Introduction

The expectations hypothesis (EH) of the term structure implies that the yield spread between the long rate and short rate is an optimal predictor of future changes in short rates over the life of the ‘long bond’.

There is a great deal of evidence on the EH for the US, based on the Campbell and Shiller (1991) VAR methodology using monthly data on spot rates (e.g. Hardouvelis, 1994). In general, for a wide variety of maturities from 1 to 12 months and for 2, 3, 4, ... 10-year maturities, Campbell and Shiller (1991) reject the EH. The (short) interest rate spread does not predict the direction of changes in the long-term interest rate that is consistent with the EH, and future changes in short-rates are not often closely correlated with the long-short spread (Campbell and Shiller, 1991).

Kugler (1988) using US, German and Swiss monthly data on one and three month Euromarket deposit rates found support for the EH only on German data (for the period of March 1974 to August 1986). Similarly, Engsted (1994) using Danish money market rates and Engsted and Tanggaard (1994) for longer maturity bonds find considerable support for the EH when the variation in interest rates is relatively large, such as in the post-1992 ERM ‘crisis period’. This is to be expected following the analysis of Mankiw and Miron (1986), for if interest rate stabilisation results in random walk behaviour for short rates, then the expected change in short rates is zero and the spread has no predictive power for future short rates (See also Rudebusch, 1995).

Using the Campbell-Shiller VAR methodology on data at the short end of the maturity spectrum (i.e. up to one year) Cuthbertson (1996) finds reasonable support for the EH on UK data. This is in contrast to Taylor (1992) who uses maturities of 5, 10 and 15 years and finds strongly against the EH (see also MacDonald and Speight, 1991). Two recent studies have tested the EH using Irish data. Cuthbertson and Bredin (2000 and 2001) using a VAR methodology find evidence in support of the EH using a range of interest rate maturities.

The main aim of the paper is to present results based on alternative methods and to test the validity of these procedures. Although the theory suggests that the long-short spread should have some predictive power for the future change in the long-short rate, ordinary least squares (OLS) estimation of the actual change in the short rate on the lagged yield spread plus a constant, results in coefficients which are often both the wrong size and sign (Mankiw and Miron, 1986). However, empirical evidence has found that, when the same relationship is estimated using instrumental variables (IV) to allow for errors in variables or a random error in the term structure relationship, the rejection of the model is less conclusive. I will also estimate this model using generalised methods of moments (GMM) to account for any heteroscedasticity in the error term or serial correlation due to the use of overlapping errors. The model is also estimated the other way around, i.e. the yield spread is regressed on the expected future changes in the short rate, where the actual future change in the short rate proxy the expected change.

In the final section of the paper, I present the results from a number of Monte Carlo (MC) experiments. The experiments focus on the fact that the alternative single equation tests based on the regression of the change in the short-term rate on the lagged spread are prone to severe over-rejection of the EH, even when it is true. However tests of the spread on the first difference of the short-rate reject at the correct rate.

2 Theoretical Review

2.1 Traditional Single Equation Tests

The Expectations hypothesis (EH) of the term structure posits that the return on an n-period bond Rt(n) is determined solely by expectations of (current and) future rates on a set of m-period assets rt(m) where n > m. Using the continuously compounded spot rates the ‘fundamental term structure’ relationship is:

Rt(n) = ( / )1 0 1 k i k = −

∑

where k = n/m is an integer. Reparameterising (1) enables the spread to be interpreted as the optimal predictor of future changes in short rates, rt(m):

St(n,m) = Et i k = −

∑

where St(n,m) = (Rt(n) - rt(m)) is the yield spread. For example, if m = 1 the above equation states that the spread between an n-period bond and a one-period bond equals the expectation of the change in one-period rates over the n-period horizon. The

Perfect foresight spread is the spread that would be predicted by agents if they had perfect foresight about future movements in interest rates. A testable implication of equation 2 is that the spread Granger causes future changes in short rates.2 Assuming that (Rt(n) , rt(m)) are I(1), then ∆ rt(m) is I(0), which from equation 2 implies that the spread St(n,m) should also be I(0). and therefore ( Rt(n) , rt(m) ) should be co-integrated with a co-integrating parameter of unity.3

If we now add the assumption of rational expectations (RE):

rt+im(m) = Etrt+im(m) + εt+im (3) we obtain the following single equation test of the null of the ‘expectations hypothesis plus rational expectations‘, EH + RE:

PFSt(n,m) = α + βSt(n,m) + γΩt + ε*t (4)

H0 : α = γ = 0, β=1

2 _{Strictly, failure of Granger causality does not constitute a rejection of the EH, but a failure to confirm}

where εεεε*t is a moving average error of order (n-m-1) consisting of a weighted sum of future values of εεεεt+im . Under RE, εεεε*t is independent of information at time t, ΩΩΩΩt , and in particular is independent of the yield spread. If there is a constant term premia or if there are differential yet constant transactions costs (between investing ‘long‘ and in a series of rolled-over short-term investments) then α≠ 0. Under RE the right hand side variables in equation 4 are independent of ε*t and hence we do not require IV estimation. However a GMM estimator is employed to correct the covariance matrix for the moving-average error of order (n-m-1) and possible heteroscedasticity (Hansen, 1982; Newey and West, 1987).

2.2 Alternative Single Equation Tests

In this section a number of alternative single equation tests will be described. I focus on the 6 month – 3 month combination. Under the EH the 6 month rate is equal to an average of current plus expected future 3 month rates plus a term premium.

θ

+

+

=

(

1

/

2

)[

]

r

E

r

R

While the 6 month - 3 month spread is equal to a weighted average of the expected change in future 3 month rates.

3 _{Strictly, for this to hold, forecast errors and any term premia must also be I(0). Shea (1992) examines}

the possibility of multiple co-integrating vectors, an issue not explored here since we concentrate on tests of bilateral relationships.

θ

+

∆

=

)

2

/

1

(

E

r

S

₍₆₎

One can see that an appropriate test would be, the regression of the change in the short rate on the lagged yield spread and a constant (Mankiw and Miron, 1986). Taking the inverse of equation 6 and from theory the short rate should differ from its predicted value only by a forecast error, which would be orthogonal to all information at t or earlier, yields; 3 ) 3 , 6 ( ) 3 ( 3 3

2

2

=

−

+

+

∆

r

θ

S

η

₍₇₎

where ηt+3 is the forecast error.

The EH implies that the coefficient on ∆Rt+3 should be insignificantly different from 1/2, in equation 6, while the coefficient on St, in equation 7, should be insignificantly different from 2. Empirical evidence would suggest that the EH does not hold; OLS estimation yields coefficients that are of the wrong magnitude and sometimes the wrong sign (Shiller, 1990).

There are two possible explanations for the empirical rejection of the theory. The first assumes that market expectations are rational but that the information contained in the in the term structure is affected by non-stationary risk premia. The second explanation assumes that risk premia are stationary, but that market expectations are not strictly rational and so long rates tend to overreact to future short rates. I will focus on the weaker form of the EH where we have a term premium that contains elements which vary randomly over time, independent of the short-term rates. Equation 5 is replaced by;

t t t t t

r

E

r

R

=

(

1

/

2

)[

+

]

+

θ

+

ε

₍₈₎

and equation 6 by t t t t

E

r

S

=

(

1

/

2

)

∆

+

θ

+

ε

The εt is a zero-mean random term and is uncorrelated with rt+3. The εt may be thought of as representing a time varying term premium or some other error.

Equation 7 may be re-written as;

t t t t

S

r

2

θ

2

η

2

ε

=

−

+

+

+

∆

3. E

E

3.1 Evidence based on the Traditional Single Equation Tests

Campbell and Shiller (1991) use a wide variety of maturities from 1 to 12 months and for 2,3,4 ... 10-years to test the EH on monthly data from January 1952 to February 1987. For the perfect foresight spread equations Campbell and Shiller find slope coefficients ranging between 0 and 0.5 for maturities up to 2 years and for maturities greater than 2 years the slope coefficients increase to around 1. Overall they find little support for the EH at the short end, but do find some support at the long end of the maturity spectrum.

However for my purposes the studies carried out using UK data will prove important for comparison purposes4. Hurn, Moody and Muscatelli (1995) also test the

4

Although the Sterling/Irish pound link has been broken, with Ireland joining the ERM in March 1979, the two countries financial systems are still very closely linked, Walsh (1996).

term structure of interest rates at the short end of the maturity spectrum using UK LIBOR for 1, 3, 6, and 12 month maturities. In contrast to Cuthbertson (1996), the authors use monthly data over a longer sample period covering January 1975 to December 1991. The findings from the perfect foresight spread regression are supportive of the EH, with slope coefficients ranging between 0.816 for the S(12,3) months spread and 1.168 for the S(3,1) months spread combination.

Two recent studies give evidence in favour of the EH using UK data, Cuthbertson (1996) and Cuthbertson, Hayes and Nitzsche (1996). Cuthbertson (1996) uses London Interbank (offer) rates (LIBOR) with maturities of 7 days, 1, 3, 6, and 12 month to test the EH at the short end of the maturity spectrum. The data set is sample weekly beginning on the 2nd Thursday in January 1981 and ending on the 2nd Thursday of February 1992. The perfect foresight spread equations yield evidence in favour of the economic theory. The null H0 : β=1 (given γ = 0) is not rejected in all cases except that for the 4-week/1-week spread.

Cuthbertson, Hayes and Nitzsche (1996) using UK Certificates of Deposit (CD) rates for maturities of 1, 3, 6, 9 and 12 months also find evidence in favour of the EH + RE. The data set used in the study is sampled weekly and covers the period from the 1st of October 1975 to 14th of October 1992 5. In all cases the authors do not reject the null of H0: β=1, or that information, available at t or earlier does not incrementally add to the predictions of future interest rates. Therefore the results are consistent with the EH + RE and previous empirical evidence on UK rates, Cuthbertson (1996)6.

5 _{The only exception here is the 9 month CD rate which ends on the 27}th _{of January 1988.} 6

Cuthbertson and Bredin (2000) also report the perfect foresight spread results using Irish data, but for comparison we also report them in this paper.

3.2 Evidence based on the Alternative Single Equation Tests

Mankiw and Miron (1986) run an OLS regression of the change in the short-run rate ∆Rt on the lagged yield spread St-1 and a constant over a number of different sample sizes. The total sample period runs from 1890 – 1958, however the authors take into account the different monetary regimes over the total sample period. In all the predictive power of the spread is tested over 4 different regimes for the (6m, 3m) maturity combination using quarterly data7.

1

1

)

(

)

(

r

−

r

=

α

+

β

R

−

r

+

v

Initially Mankiw and Miron (1986) test the predictive power of the spread over a relatively recent sample period, 1959 – 1979. The authors find a coefficient on the spread that is insignificantly different from zero and significantly different from the theoretical value of 2. The authors also find an adjusted R-squared of 0.01, which implies that the spread has negligible predictive power. Overall the results are not supportive of the expectations hypothesis and suggest that the slope of the yield curve does not contain information regarding the future path of the short rate.

The results for the various sub-samples offer some interesting comparisons. For the sub-samples over the period 1915 – 1958 the results are remarkably similar to the recent sample and the spread is not significantly different from zero. However for the sample period 1890 – 1914, the results are in marked contrast and, although the coefficient on the spread is still statistically different from its theoretical value, it is three times larger that that for the recent sample. The predictive power is also much

7 _{The 4 sub-samples for each of the different regimes are the following. 1890Q4 – 1914Q4 which ends}

with the founding of the Federal Reserve System. 1915Q1 – 1993Q4 ends with both the introduction of the New Deal banking reforms and the approximate end of the gold standard and approximate beginning of interest rate begging. 1934Q1 – 1951Q1which ends with the Fed no longer pegging interest rates and

greater, the adjusted R2 is 0.40 compared to 0.01 for the recent sample. Therefore the sample period 1890-1914 confirm that expectations are an important factor of fluctuations in the yield curve.

Two recent studies include Sola and Driffill (1994) and Driffill, Psaradakis and Sola (1997). Sola and Driffill (1994) test the expectations hypothesis for the 3 and 6 month combination using quarterly data for US treasury bills for the sample 1962Q1 – 1987Q3. Given the change in the Federal Reserves operating procedure towards monetary base control in the late 70’s the authors initially test the theory from 1962Q2 – 1979Q3 and so avoid any possible rejection based on the shift in regime8. The authors initially estimate equation 7 using OLS and find the coefficient on the lagged spread is significantly different from the theoretical value, 2. However as has been discussed above, the failure of equation 7 may be due to time varying term premia, fads, measurement errors, or other random deviations from the pure expectations hypothesis.

The authors re-estimate the model using instrumental variables (IV), using St-i, i= 2,…,4, and ∆rt-i, i = 1,…,4 as instruments. The results do not find evidence against the weak version of the EH. Sola and Driffill (1994) also estimate equation 10 using

St-i, i= 1,…,4, and ∆rt-i, i = 0,…,4 as instruments. The coefficient on the change on the short rate is not statistically different from the theoretically correct value of 0.5. The authors conclude that equation 10 is the correct model and there is a measurement or other error in equation 7. In a Monte Carlo study Driffill, Psaradakis and Sola (1998) confirm that the results found in real data are likely to emerge when equation 10 is the correct model. The authors also estimate the models over the full sample period 1962Q2 – 1987Q3 and find unsurprisingly that the theory is rejected when there is no allowance made for the regime switch.

finally 1951Q2 – 1958Q4 ending at the time when the active market in 3 and 6 month Treasury bills begins.

8 _{There are 2 possible reasons why the theory may be rejected given the switch in regime. There may be}

a break in the time series properties of the data given the regime switching. Secondly there may be a perception of future shifts which may affect the behaviour of market participants, .i.e. ‘peso problem’.

Driffill, Zacharias and Sola (1997) test the expectations hypothesis for UK and US 1 and 3 month combinations using monthly data. The sample period for UK covers the period 1975M2 - 1994M12 and 1982M12 - 1991M2 for the US. The authors also analyse the theory for the UK using a shorter sample period, starting 1982M11, given the possible structural breaks in the preceding period9. The authors recognise the possible measurement or other error in the relationship between the spread and the expected future change in the short rate and so focus on;

Again the authors run the single equation tests using IV, and find that the results are well within the statistically significance level. Also of note is the fact that non-rejection of the theory was found in both samples for the UK.

4.0 E

R

4.1 T

S

C

S

R

The regression of the perfect foresight spread, PFSt(n,m) on the actual spread St(n,m) and the limited information set Ht (consisting of lags of St(n,m) and ∆rt(m)) are shown in table 1. Under the null hypothesis of PEH + RE, we expect H0 : α = γ = 0,

β=1. The method estimation is GMM with a correction for heteroscedasticity and moving average errors using the Newey-West (1987) declining weights.

9

There were a number of causes for these structural changes; the change in monetary policy regime that accompanied the Medium Term Financial Strategy, and the elimination of all foreign exchange controls.

10 _{Unit root tests on the individual series R}

t and St(n,m) indicate that we cannot reject the null hypothesis

that changes in short rates ∆rt (m)

and the yield spread St (n,m)

are I(0). A detailed discussion of the unit root properties of the data is discussed in Cuthbertson and Bredin (2000), where a similar data set is used.

S

I first run the model including the information set and test, H2: γ = 0. As can be

seen from table 1, I cannot reject the null and so is consistent with the theory. In all cases I do not reject the null of H0 : β=1 or that information, available at time t or

earlier does not incrementally add to the predictions of future interest rates. This is the case for the each of the chosen interest rate maturity combinations, at the short end of the spectrum. The results therefore do not reject the EH + RE.

4.2 A

S

E

T

R

I now focus on alternative single equation tests, described earlier in the paper. I focus on the 6 month and the 3 month interest rate combination, as a comparison with Sola and Driffill (1994). The method of estimation includes IV and GMM. Based on previous empirical evidence, Shiller (1990), OLS estimation yields coefficients that are often both the wrong sign and magnitude. Therefore I focus on the errors in variables, IV estimation and GMM. I also run a Wald test of the restriction implied by the EH.

I first consider equation 6, where the 6 month – 3 month spread (St) is

regressed on the change (3 month) in the future 3 month short rate (∆rt+3) and a

Regressing St on a constant and ∆rt+3, using St-i, i = 1, …, 4 and ∆rt-i, i = 0, …,

4, as instruments yields;

Instrumental Variables (IV)

St = 0.0005 + 0.37∆∆∆∆Rt+3 (12)

[0.0005] + [0.05]

Instruments: St-1, St-2, St-3, St-4, ∆∆∆∆rt, ∆∆∆∆rt-3, ∆∆∆∆rt-6, ∆∆∆∆rt-9, ∆∆∆∆rt-12.

Sample Size = 147, Standard Error = 0.005, R2 = 0.31 AR[χ2(12)] = 45.77, HET[χ2(1)] = 0.31

Wald Test of the Expectations Hypothesis Restriction: χ2(1) = 5.41

I also give a summary of the diagnostic tests for each regression. The AR test is a test for serial correlation up to order 12, while the HET test is a test of unconditional heterscedasticity. As can be seen from the above equation the estimated coefficient on the change in the short rate appears to be different from the theoretically correct value. The Wald test result rejects the restriction implied by the EH. As has already been discussed these results have been taken as evidence against the EH.

Given that ε*t may have a moving average error and be possibly

heteroscedastic, I also estimate the model using a GMM estimator to correct the covariance matrix for the moving-average error of order (n-m-1) and possible heteroscedasticity (Hansen, 1982; Newey and West, 1987).

Generalised Method of Moments (GMM)

St = 0.001 + 0.39∆∆∆∆rt+3 (13)

[0.0004] + [0.06]

Instruments: St-1, St-2, St-3, St-4, ∆∆∆∆rt, ∆∆∆∆rt-3, ∆∆∆∆rt-6, ∆∆∆∆rt-9, ∆∆∆∆rt-12.

Sample Size = 147, Standard Error = 0.006, R2 = 0.31 AR[χ2(12)] = 92.64, HET[χ2(1)] = 41.11

Wald Test of the Expectations Hypothesis Restriction: χ2(1) = 4.08

Overall the results are quite similar to those using the IV estimation. As can be seen from both the regression results and the Wald test, the estimated coefficient on the change in the short rate appears to be different from the theoretically correct value.

I also estimate equation 7, by regressing the change (3 month) in the 3 month short rate (∆Rt) on the 3 month lagged spread (St-3). Again, I estimate using both IV

and GMM estimation.

Instrumental Variables (IV)

∆∆∆∆rt = - 0.001 + 3.02St-3 (14)

[0.001] + [0.71]

Instruments: ∆∆∆∆rt-1, ∆∆∆∆rt-2, ∆∆∆∆rt-3, St-6, St-9, St-12.

Sample Size = 150, Standard Error = 0.01, R2 = 0.31 AR[χ2(12)] = 87.23, HET[χ2(1)] = 2.91

As can be seen from the above regression results, the Wald test overwhelmingly rejects the restriction implied by the EH. I also report the results with a correction for possible heteroscedasticity and moving average error.

Generalised Method of Moments (GMM)

∆∆∆∆rt = - 0.002 + 3.05St-3 (15)

[0.001] + [0.78]

Instruments: ∆∆∆∆rt-1, ∆∆∆∆rt-2, ∆∆∆∆rt-3, St-6, St-9, St-12.

Sample Size = 150, Standard Error = 0.01, R2 = 0.31 AR[χ2(12)] = 92.64, HET[χ2(1)] = 41.11

Wald Test of the Expectations Hypothesis Restriction: χ2(1) = 298.30

Overall the results are quite similar to those using the IV estimation. As can be seen from both the regression results and the Wald test, the estimated coefficient on lagged spread is significantly different from the theoretically correct value.

5.0 M

C

E

In the previous section of this paper, I used two alternative methods to test the weak version of the EH, considering time varying term premium, using single equation estimation. I now focus on the findings that the tests based on the regression of the change in the short-term rate on the lagged spread is prone to severe over-rejection of the EH. However tests of the spread on the first difference of the short-rate reject at the correct rate. I will use Monte Carlo (MC) experiments to show this point. This fact has been briefly discussed in the section dealing with the empirical results on these alternative single equation tests and can be seen at first hand from the reported results

in the previous section. The procedure used will be to estimate both methods of the single equation tests for 3 alternative samples; T = 100, 200 and 50011. A 1000 series of these regressions will be produced. I set up the experiments based on the Irish data set already analysed here. In the experiments I take the long rate as the 2 period rate (6 month), and the short rate as the 1 period rate (3 month). The generating process is determined by the following equations;

∑

As has been mentioned the simulations are carried out based on the previously used Irish data set. Based on the data set, θ = 0.0077, µ = 0.0088, b1 = 1.11, b2 = -0.47, b3 = 0.24, σ_ε = 0.011, σu = 0.012. As has been mentioned 1000 series of regressions will be generated, and the pseudo-random deviates ut and εt will be

obtained using the RNDN function in GAUSS. As a direct comparison with the previous section, estimation will be by IV and GMM.

Model 1: The DS Test

∆rt = α1 + β1St-1 + e1t Instruments: St-2, St-3, St-4.

β1=2

Model 2: The SD Test

St = α2 + β2∆rt+1 + e2t

Instruments: ∆∆∆∆rt , ∆∆∆∆rt-1 , ∆∆∆∆rt-2. β2=0.5

11

The sample size will be T + 50, in each replication. Then the first 50 data points will be dropped in order to take account of start-up effects.

In table 2, I present the results for the mean bias for models. As can be clearly seen from the table of results, model 1 is prone to a large amount of over-rejection of the EH, even when it is true. This is the case for both the IV and the GMM estimator. As can be seen the bias for model 1 continues to be sizeable, even with a larger sample size. On the other hand, model 2's bias is much smaller and is not significant in any of the cases. In table 3, I report the findings for the test rejection frequencies for the 2 models. Again the results are consistent with that in table 2. The results are based on both the t test and the Wald test at the 5% significance level, that the beta value is equal to its theoretical value. As can be seen the fraction that reject for model 1, is much greater than that for model 2. Therefore, model 1 rejects the EH even when it is true. The results reported in this section are consistent with those reported in a similar study by Driffill, Psaradkis and Sola (1998), which uses MC experiments12.

6.0 C

This study looks at a number of traditional and alternative tests of the EH and uses MC experiments to test the validity. We have focused on tests of whether the spread predicts future changes in short rates at a number of interest rate maturities. In all cases we do not reject the null of H0 : β=1 or that information, available at time t or

earlier does not incrementally add to the predictions of future interest rates. This is the case for the each of the chosen interest rate maturity combinations, at the short end of the spectrum. The results therefore do not reject the EH + RE.

This paper has also looked at some alternative single equation tests of the EH. Focusing in particular on the 6 month and 3 month maturities, I initially test the model using OLS. However given the previous evidence, that OLS estimation of the actual change in the short rate on the lagged yield spread plus a constant, results in coefficients which are often both the wrong size and sign (Mankiw and Miron, 1986), I also use IV and GMM estimation. Although both regressions reject the EH, the

regression of the change in the short-term rate on the lagged spread appears to reject to a much larger degree.

Finally, I also report the results for the MC experiments which show that the single equation tests based on the regression of the change in the short-term rate on the lagged spread are prone to severe over-rejection of the EH. However the tests of the spread on the first difference of the short-rate reject at the correct rate. These findings are consistent with those from Driffill, Psaradkis and Sola (1998) using US data.

data.

R

Campbell, J.Y. and Shiller, R.J. (1991). ‘Yield Spreads and Interest Rate Movements : a Birds Eye View’, Review of Economic Studies, Vol. 58, pp.495-514.

Cuthbertson, K. (1996) ‘The Expectations Hypothesis of the Term Structure: The UK Interbank Market’, The Economic Journal, Vol.106, No.436, pp.578-592.

Cuthbertson, K. and Bredin, D. (2000) ‘The Expectations Hypothesis of the Term Structure: The Case of Ireland’, The Economic and Social Review, Vol. 31, No. 3, pp. 267-281.

Cuthbertson, K. and Bredin, D. (2001) ‘Risk Premia and Long Rates in Ireland’, Journal of Forecasting (forthcoming).

Cuthbertson, K., Hayes, S. and Nitzsche, D. (1996) ‘The Behaviour of Certificates of Deposits Rates in the UK’, Oxford Economic Papers, Vol. 48, pp.397-414.

Driffill, J., Psaradakis, Z. and Sola, M. (1997) ‘A Reconciliation of Paradoxical Empirical Results on the Expectations Model of the Term Structure’, Oxford Bulletin of Economics and Statistics, 59, pp. 29-42.

Driffill, J., Psaradakis, Z. and Sola, M. (1998) ‘Testing the Expectations Hypothesis of the Term Structure Using Instrumental Variables’, International Journal of Finance and Economics, 3, pp. 321-325.

Engsted, T. (1994). ‘The Predictive of the Money Market Term Structure’ Aarhus School of Business, Denmark (mimeo).

Engsted, T. and Tanggaard, C. (1994) ‘A Cointegration Analysis of Danish Zero-Coupon Bond Yields’, Applied Financial Economics, Vol. 24, pp.265-78.

Hansen, L.P. (1982). ‘Large Sample Properties of Generalised Method of Moments Estimators’, Econometrica, Vol. 12, pp. 1029-52.

Hardouvelis, G.A. (1994). ‘The Term Structure Spread and Future Changes in Long and Short-Rates in the G7 Countries’, Journal of Monetary Economics, Vol. 33, pp.255-83.

Hurn, A.S., Moody, T. and Muscatelli, V.A. (1995) ‘The Term Structure of Interest Rates in the London Interbank Market’, Oxford Economic Papers, Vol. 47, No. 3, pp. 418-36.

Kugler, P. (1988) ‘An Empirical Note on the Term Structure and Interest Rate Stabilization Policies‘, Quarterly Journal of Economics, Vol. 103, No. 4, pp. 789-92.

MacDonald, R. and Speight, A.E.H. (1991) ‘The Term Structure of Interest Rates Under Rational Expectations : Some International Evidence‘, Applied Financial Economics, Vol.1 pp.221-21.

Mankiw, N.G. and Miron, J.A. (1986) ‘The Changing Behaviour of the Term Structure of Interest Rates‘, Quarterly Journal of Economics, Vol. 101, pp. 221-28.

Newey, W.K. and West, K.D. (1987). ‘A Simple Positive Definite Heteroscedasticity and Autocorrelation Consistent Covariance Matrix’, Econometrica, Vol. 55, pp.703-8. Shea, G.S. (1992) ‘Benchmarking the Expectations Hypothesis of the Term Structure: An Analysis of Cointegration Vectors’, Journal of Business and Economics Statistics, July, pp.347-65.

Shiller, R.J. (1990) The Term Structure of Interest Rates’, in Freidman, B.M and Hahn, F.H. (Eds.), Handbook of Monetary Economics, Amsterdam, Horth Holland.

Sola, M. and Driffill, J. (1994) ‘Testing the Term Structure of Interest Rates using Stationary Vector Autoregression with Regime Switching’, Journal of Economic Dynamics and Control, Vol. 18, pp. 601-628.

Taylor, M.P. (1992). ‘Modelling the Yield Curve’, The Economic Journal, Vol. 102, No.412, pp.524-37.

Walsh, B. (1996) ‘The Real Exchange Rate, Fiscal Policy and the Current Account: Interpreting Recent Irish Experience’, Centre for Economic Research WP96/10.

Table 1 :

Does the Spread Predict Future Changes in Short-Rates ?

Regression : PFS

=

αααα

+

ββββ

S

+

γγγγΩ

Ω

Ω

Ω

Spread (n,m)

Coefficients

Wald

Test

α

s.e.(

α

)

β

s.e.(

β

)

H

:

β

=1

[p-value]

H

:

α

=0,

β

=1

[p-value]

H

:

γ

=0

[p-value]

(6,3) Month

-0.0006

(0.0008)

1.02

(0.23)

0.007

[0.93]

0.007

[0.93]

2.88

[0.09]

(3,1) Month

-0.001

(0.001)

0.87

(0.21)

0.38

[0.54]

0.39

[0.53]

3.75

[0.06]

(6,1) Month

-0.001

(0.001)

1.04

(0.15)

0.09

[0.77]

0.08

[0.78]

0.32

[0.58]

The regression coefficients reported in columns 2 and 3 are from the regression with γ = 0 imposed. The method of estimation is GMM with a correction for heteroscedasticity and moving average errors using the Newey-West (1987) declining weights. The last 3 columns report Wald statistics and marginal significance levels for the null hypothesis stated. For H0 :γ = 0 the reported results are for an information set which includes 4 lags of the change in the interest rates and the interest rate spread. The

null H0:β=1, is conditional on γ=0 while the null H1:α=0, β=1 is also conditional on γ=0.

Table 2 :

Monte Carlo Bias

Mean Bias of the IV Estimators

T =

Model 1

Model 2

100

-1.896

(0.0396)

-0.0370

(0.0709)

200

-1.844

(0.0398)

-0.0172

(0.0702)

500

-1.830

(0.0336)

-0.0095

(0.0667)

Mean Bias of the GMM Estimators

T =

Model 1

Model 2

100

-1.888

(0.0363)

-0.0356

(0.069)

200

-1.868

(0.0338)

-0.0197

(0.0675)

500

-1.785

(0.0365)

-0.0119

(0.0669)

Table 3 :

Test Rejection Frequencies:

Test Rejection Frequencies – IV Estimators (T-Test)

T =

Model 1

Model 2

100

0.540 0.063

200

0.548 0.054

500

0.524 0.051

Test Rejection Frequencies – GMM Estimators (Wald-Test)

T =

Model 1

Model 2

100

0.580 0.069

200

0.575 0.065

500

0.540 0.058

Notes:

The reported results give the fraction of MC replications that fail the t-test and the Wald test at the 5% significance level. Both the t-test and the Wald test, test whether the value for the beta for both model 1 and model 2 are equal to the theoretical value.